Supplementary Information : Cross-Platform Comparison of Arbitrary Quantum States

The parameters MU and MS can be optimized through minimizing the statistical error with grid search 1, 2 or using the importance sampling with partial information on the quantum state 3. Both approaches require prior knowledge or simulation of the target state. Here, we devise a greedy method for sampling the unitary operation U that reduces the statistical error without prior knowledge of the target state. The statistical error as a function of MU converges faster than uniformly sampling the unitary operation when the number of shots MS 2 , where N is the number of qubits. Therefore, the greedy method is particularly useful for the 5and 7-qubit experiments. In this section, we demonstrate the comparison between the greedy method and random method for 5-qubit GHZ state.

We compare the two different methods of sampling the random unitary U : the randomized sampling and the greedy method. Using these two methods, we evaluate the fidelity between the states prepared on the UMD 1 system and the IBM 1 system, by sampling subsets of various sizes M U from the full state tomography measurements. Figure S1 shows the error of the fidelity estimation between UMD 1 and IBM 1 as a function of M U for M S = 2000. We see that the greedy method outperforms the random method in this regime.
To further characterize the performance of the greedy method, we perform numerical simulation through Pauli basis measurements. Specifically, to generate the random Pauli measurement using the greedy method, we define a set of unitary operators S = {H, HSH, I} to perform measurement in the x, y, and z basis. Using Algorithm. 1, we can sample the greedy random unitary operators.
We compare the measurement results for the greedy method and the random sampling method  Figure S2: The number of Pauli-string operators that can be predicted through the set of measurements (N obs ) as function of M U . In the large M U 3 N limit, the measurement is equivalent to full state tomography and therefore, all the Pauli-string operator can be predicted (N obs /4 N = 1).

Random Greedy
However, in the regime M U < 3 N , we show that Greedy method can predict more observables than the random method.
in the regime M S 2 N . First, we compute the number of Pauli-string operators that can be predicted 2 through the set of measurements (N obs ). In Fig. S2, we present the N obs as function of M U , normalized by the total number of Pauli string operators 4 N . We see that the greedy method can predict more observables than the random method in the regime, M U < 3 N . However, when M U 3 N , both greedy and random method can predict all 4 N Pauli string observables. Therefore, the shot noise is the dominant error source.
Second, we perform the simulation for the prediction of the linear observable tr(Oρ) and

S3 SWAP overhead for quantum volume circuits
Two-qubit gates on non-nearest-neighbor pairs are not directly available on superconducting quantum computers. To realize such non-nearest-neighbor two-qubit gates effectively, SWAP gates are necessary. Note each SWAP gate consists of three CNOT gates. Thus, when used, non-trivial degradation to the overall fidelity of the computational output state is incurred.
Optimizing the so-called qubit routing can effectively decrease the number of involved nonnearest-neighbor two-qubit gates in evaluating the quantum volume circuits. As the number of layers d increases though, the non-nearest-neighbor two-qubit gate becomes unavoidable. In Fig.  S5 we show the mean value of two-qubit gates used to implement quantum volume circuits of d layers on different platforms. The value grows linearly with d.

S4 Quantum systems
In this section we detail the quantum systems used in this study.

IBM Quantum Experience
We use IBM Quantum Experience service to access several of their superconducting quantum

TI UMD (UMD 2)
The second trapped-ion quantum computer system at Maryland is part of the TIQC (Trapped Ion Quantum Computation) team. This quantum computer supports up to nine qubits made of a single chain of 171 Yb + ions trapped in a linear Paul trap with blade electrodes 9 . Typical single-and two-qubit gate fidelities are 99.5(2)% and 98−99%, respectively. On this platform, we compile the quantum volume to its native gate set through KAK decomposition. We apply SPAM correction to mitigate the detection noise assuming that the preparation noise is negligible.

IonQ (IonQ 1 and IonQ 2)
The commercial trapped-ion quantum systems used by IonQ contain eleven fully connected qubits in a single chain of 171 Yb + ions trapped in a linear Paul trap with surface electrodes 9 . The single-qubit fidelities are 99.7% for both systems at the time of measurement, while two-qubit fidelities are 95 − 96% and 96 − 97% for IonQ 1 and IonQ 2 respectively. On this platform, we apply the technique described in Ref. 10 to optimize the circuit. No SPAM correction was applied in post-processing.

S5 Intra-Technology Similarity through Principal component analysis
We consider quantum states reconstructed from the shadow tomography measurement T is the number of classical shadows. After averaging over T classical shadows, we decompose the density matrix defined in Eq. (S1) as a linear combination of Pauli string operators, ρ = 4 n −1 k=0 v k P k , where P k is the Pauli string operator and v k = 1 2 n tr(ρP k ). Therefore, a density matrix ρ can be represented by a 4 n -dimensional vector v = 4 n −1 k=0 v kêk . We define the vector v as the feature vector for the principal component analysis.
In the noiseless limit, the vector that represents the target state |ψ is v t = 4 n −1 k=0 ψ|P k |ψ ê k .

S6 Measurement error mitigation
In this section, we introduce the measurement error of and the error mitigation technique used for the IBM superconducting qubits. Measurement error is one of the dominant errors in the superconducting qubit devices. A measurement error manifests itself as either a |0 state being read as a |1 state or vice versa. For a quantum computer with n qubits, the measurement error can be fully described by a 2 n × 2 n measurement error matrix M . The matrix element M [s, s ] is the probability of measuring outcome s when the quantum computer is in state |s . Therefore, if we take the probability vector P ideal describing the ideal measurement results for a given circuit, applying the measurement matrix M gives a good approximation of the results when measurement noise is present. In particular, In order to approximate the P ideal , we perform an optimization to minimize the cost function subject to constraints 0 ≤ P ideal (s) ≤ 1 and s P ideal (s) = 1.
For the devices IBM 2 and IBM 3 with seven qubits, we measure the measurement error matrix M by initializing the qubits in all the 2 7 possible bit strings and measure each state with 2048 shots. The measurement error matrix for IBM 2 is shown in Fig. S7(a). The dominant measurement error is the single qubit flip. In particular, the error rate for measuring |1 when the state is |0 ranges from 1% to 8%. The error rate for measuring |0 if the state is |1 ranges from